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Extra resources for The finite element method : basic concepts and applications
Sample text
Derive Eq. 32) from Eq. 31). 2 and verify the solution. Show that if the expressions for (i) given by Eq. 20) are used in Eq. 19), then ai = Ti. Find the weak formulation of Eq. 13 with L = l m, k = 200 w/mK, Q = 100 W/m3, h = 150 W/m2 K, T∞ = 100°C. REFERENCES Dawe, D. J. (1984). Matrix and Finite Element Displacement Analysis of Structures. Oxford: Clarendon Press. Fletcher, C. A. J. (1984). Computational Galerkin Methods. Berlin: Springer-Verlag. Huebner, K. , Thornton, E. , and Byrom, T. G.
For a prescribed heat flux q at x = 0, the boundary term in Eq. 77) now renders a contribution at x = x2 k −1 , which is K dT |x =0 = − q dx and the total contribution of the element becomes ⎡ 14 − 16 2 ⎤ ⎡T1 ⎤ ⎡1 ⎤ ⎥⎢ ⎥ ⎢ ⎥ K⎢ ⎢ −16 32 − 16 ⎥ ⎢T2 ⎥ − q ⎢0 ⎥ = 0 6 ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ 2 − 16 14 ⎥ ⎢T3 ⎥ ⎢0 ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ as we expected. 4 Let us now evaluate the integral ∫ WQdx ei where Q is a constant internal heat source, for both linear and quadratic elements, using natural coordinates. 82) where we used Eqs.
2 ⎢ h ( ei ) ⎢ (1 + 1 + 1)! = 2 ⎢ 1! 1! ⎢2 ⎢⎣ (1 + 1 + 1)! = 1! 1! ⎤ (1 + 1 + 1)! ⎥ ⎥ 0! 2! ⎥ ⎥ 2 (1 + 0 + 2)! 71) h ( e1 ) ⎡2 1⎤ ⎥ ⎢ 6 ⎢1 2 ⎥ ⎦ ⎣ Notice that this is a general relation for any element involving the product [ N ]T [ N ]. 72) 1 ⎡ 1 − 1⎤ ⎥ ⎢ h ( ei ) ⎢ −1 1 ⎥ ⎦ ⎣ which is the familiar diffusion integral. We can easily utilize the natural coordinate system for higher-order elements, in particular, the quadratic element. 74) pjwstk|402064|1435431860 Notice that this expression reduces to Eq.